can you typically compare lengths in compass and straightedge constructions?

Question

Historically, has it been considered valid in compass and straightedge constructions to, given two line segments, decide which is longer? It isn't hard to see that this problem is equivalent to the problem of, given 3 co-linear points, choosing the one which lies in the middle. Does assuming this principle actually improve one's power of construction?

Answer 1

Yes, we can use compass-and-straightedge construction to reduce the question of which line segment is longer to an issue of which of three distinct co-linear points is between the other two.

This is an incidence relationship: does point $B$ belong to line segment $AC$? Much like the incidence of a point with a line, the wording of Euclid's Elements seems clearly to assume a geometer can draw that distinction.

Consider the early placement of Book I, Proposition 3:

To cut off from the greater of two given unequal straight lines a straight line equal to the less.

The proof given, a compass-and-straightedge construction, relies on the preceding Proposition 2:

To place a straight line equal to a given straight line with one end at a given point.

The alert Reader will recognize this last as enabling the comparison alluded to in the Question, by transferring the length of one given line segment to a line segment sharing an endpoint with another given line segment. Having accomplished that, the compass is used to draw a circle around the shared endpoint with radius equal to the transferred line segment length.

IF this is the lesser length, THEN the circle intersects the greater length line segment. The geometer recognizes/labels this point of intersection as a cut off.

On the other hand, should the transferred length exceed the other line segment length, then there will not be a point of intersection. In this sense the compass-and-straightedge construction given to prove Book I, Prop. 3 is an effective test to "decide which is longer".

There is of course room for skepticism about exactly what properties of compass-and-straightedge constructions are used in this test. It is fairly explicit in Book I, Postulate 3 that what the compass draws is a circle (as defined in Book I, Def. 15-16 to have all radii of equal lengths).

Missing from these early parts of Euclid's Elements is a direct postulation of (mathematical) trichotomy, i.e. that if two things are not equal, then one is greater and the other is lesser. Lack of such a postulate does not prevent Euclid from appealing to this principle later, cf. Book XII, Prop. 2 inter alia.

Answer 2

Euclid's algorithm depends on not only deciding one line segment is longer than another, but subtracting the smaller length from the larger repeatedly. As far as deciding betweenness of points on a line, Euclid is silent, assuming it is given implicilty. Axiomatization of betweenness came much later in the 19th century. Also, at that time there was a study of Euclidean constructions under various constraints. This is mentioned briefly in the Wikipedia article Compass-and-straightedge construction.